Construction of three-dimensional profiles of high aspect ratio structures using top down imaging

ABSTRACT

The methods and systems disclosed here leverage currently available reliable top down imaging techniques used by SEMs and use computational methods to synthesize accurate  3 D profiles of features of high aspect ratio structures in a device. Radial cross-sectional profiles obtained from different locations along the lateral direction at different heights/depths are stitched together to create one composite  3 D profile of the HAR feature.

TECHNICAL FIELD

Embodiments of the disclosure relate generally imaging of high aspectratio (HAR) structures, and specifically to characterization of highaspect ratio structures on a semiconductor wafer or other structureusing precision three-dimensional profiles derived from top-down imagedata.

BACKGROUND

A variety of HAR structures are routinely used in current and nextgeneration semiconductor devices. Features of a HAR structure should becharacterized well using detailed metrology to be able to tune the etchprocess parameters as the etch progresses and the aspect ratio changes.

Existing imaging approaches for metrology, such as scanning electronmicroscopy (SEM), image features of any structure, including high aspectratio (HAR) structures, in either lateral or longitudinal planes. Sincethe HAR structure is sampled only on a single plane, a significantamount of information on the feature is lost. This is akin tosignificant aliasing given that metrology off a single plane is used torepresent a 3-dimensional surface. Imperfect device characteristicsgathered from the aliased images impair effective tuning of processparameters. More recently, techniques like Small Angle X-ray Scattering(known as CD-SAXS) are being used for improving device characteristicmetrology, but X-ray based techniques do not have the robust,well-established imaging capability that SEMs have.

SUMMARY

The following is a simplified summary of the disclosure in order toprovide a basic understanding of some aspects of the disclosure. Thissummary is not an extensive overview of the disclosure. It is intendedto neither identify key or critical elements of the disclosure, nordelineate any scope of the particular implementations of the disclosureor any scope of the claims. Its sole purpose is to present some conceptsof the disclosure in a simplified form as a prelude to the more detaileddescription that is presented later.

The methods and systems disclosed here leverage currently availablereliable imaging techniques used by SEMs and use computational methodsto synthesize accurate 3D profiles of features of high aspect ratiostructures in a device. Radial cross-sectional profiles obtained fromdifferent locations along a lateral direction at differentheights/depths are stitched together to create one composite 3D profileof the HAR feature.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure will be understood more fully from the detaileddescription given below and from the accompanying drawings of variousembodiments of the disclosure.

FIG. 1 illustrates a portion of an example device structure with anetched hard mask on top of a multi-layer stack, according to anembodiment of the present disclosure;

FIG. 2 illustrates a wedge-shaped test structure prepared for verticalsampling of HAR features, according to an embodiment of the presentdisclosure;

FIG. 3 illustrates a close-up of a portion of the structure in FIG. 1,according to an embodiment of the present disclosure;

FIG. 4 illustrates a close-up of a portion of the structure in FIG. 2,according to an embodiment of the present disclosure;

FIG. 5 shows two different views obtained from SEM imaging of thewedge-shaped test structure in FIG. 2, according to an embodiment of thepresent disclosure;

FIG. 6 shows a two-dimensional shape representing a memory hole channel,according to an embodiment of the present disclosure;

FIG. 7 shows a 3D rendering of a reconstructed profile of a memory hole,according to an embodiment of the present disclosure;

FIGS. 8A-8D show four different radial cross-sections of a memory holethat require increasingly higher order mathematical fits according to anembodiment of the present disclosure;

FIG. 9 shows a flowchart depicting a method of reconstructing a 3Dprofile of a HAR feature, according to an embodiment of the presentdisclosure;

FIG. 10 illustrates harmonic analysis of data obtained from a singleslice after the FFT, according to an embodiment of the presentdisclosure;

FIG. 11 illustrates the spectrogram where the zeroth and first harmoniccomponents are filtered out, according to an embodiment of the presentdisclosure;

FIG. 12 shows the evolution of δx and critical dimension (CD) with depthin the z-direction, according to an embodiment of the presentdisclosure;

FIG. 13 plots the probability density of CD, according to an embodimentof the present disclosure;

FIG. 14 plots the probability density of circularity, according to anembodiment of the present disclosure;

FIG. 15 shows at a certain depth, the tilt of a memory hole variesspatially in the x-y plane, with the arrows representing both themagnitude and direction of the tilt, according to an embodiment of thepresent disclosure;

FIG. 16 shows raw data obtained from individual blobs for CD, accordingto an embodiment of the present disclosure;

FIG. 17 shows filtered data using smoothing operations (also referred toas smoothening operations), according to an embodiment of the presentdisclosure;

FIG. 18 shows both the nominal value and the confidence interval of CD,according to an embodiment of the present disclosure;

FIG. 19 shows both the nominal value and the confidence interval ofcircularity, according to an embodiment of the present disclosure;

FIG. 20 shows a system environment where embodiments of the presentdisclosure can be practiced.

DETAILED DESCRIPTION

Embodiments of the present disclosure are directed to novel,high-resolution techniques to construct three-dimensional profiles ofcharacteristic features of High Aspect Ratio (HAR) structures usingScanning Electron Microscope (SEM) images. HAR structures that areroutinely used in current and next generation semiconductor devices,display devices, photovoltaic devices, micro-electro-mechanical systems(MEMS) devices, etc. usually have aspect ratio greater than 1:10, andmore typically, in the range of 1:40 to 1:200. However, this disclosureis not limited to any specific aspect ratio. Examples of HAR structuresinclude, but are not limited to, hard masks, contact holes, channelholes, slits, etc. Specific examples include word-line contacts andword-line isolation in three dimensional NOT-AND (3DNAND) logic gatememory devices. Further examples include Dynamic Random Access Memory(DRAM) capacitors.

The HAR features should be characterized well using detailed metrologyto be able to tune process parameters as a process (such as an etchingprocess or a deposition process) progresses and the aspect ratio of HARstructures changes. For example, in an etch process, the etch ratevaries as the aspect ratio of a feature changes with time. Accuratecharacterization of HAR features enables effective tuning of the etchprocess parameters. Current approaches for HAR feature characterizationuse SEM images along a vertical (or longitudinal) section, and/ortransmission electron microscopy (TEM) images. These imaging techniquesusually provide only an image of a single planar section from which alimited number of device characterization metrics are obtained. Thisleads to aliasing in the profile reconstruction of the HAR features. Theapproach disclosed herein avoids aliasing in reconstructed images byensuring that the sampling frequency is greater than the Nyquist spatialfrequency of the features and provides rapid high resolution metrologyusing existing SEM imaging techniques.

The present method avoids the problem of aliasing by performing normaltop down imaging, and extracting radial cross-sectional geometries atdifferent heights/depths of a HAR feature from an array of structuresthat are geometrically identical on the mask design, but can varyslightly when actually fabricated on a wafer due to process-inducedspatial variation across a wafer or substrate. The radialcross-sectional profiles obtained from different locations along thelateral direction (i.e. x-direction) at different heights/depths arethen stitched together to create one composite 3D profile of the HARfeature. Specifically, embodiments of the present method use a millingtool, such as a Focused Ion Beam (FIB), to generate a test structure(also referred to as a “coupon”) to enable sampling of a processed(e.g., after etching) HAR feature in the z-direction. The test structurecan be wedge-shaped, or staircase-shaped, or any other shape that allowssampling along the direction of height/depth of the HAR feature beingcharacterized. Another variant can include ion-beam milling and SEMimaging on the same platform with images taken as the milling progressesin the vertical direction. During testing of a process, it can bedesirable to utilize sample material in an efficient manner. Therefore,it is common to divide substrate material into smaller units known ascoupons. As used herein, coupon will be understood to mean a smallersection of a substrate. A coupon can have all of the properties andfunctionality as the substrate. For example, if the substrate is asemiconductor wafer with a plurality of devices thereon, then a couponcan be a section of the wafer and may also contain a plurality of thedevices.

Advantages of the current method include, but are not limited to: (1)direct extraction of key parameters (such as critical dimension (CD),striations, and tilt) from a set of top-down SEM images of a HARfeature, (2) ability to generate metrology statistics from a large setof image data, and (3) ability to tune process parameters in areasonably short time. In an illustrative example, the combinedoperations of etching, creating a wedge-shaped test structure (e.g., byan FIB tool), and performing SEM analysis, enable a user to retune etchprocess parameters within a time window of tens of minutes. Moreover,top down SEM-based techniques have the potential to eventually bescalable to in-line monitoring. This can be done by either ion-beammilling only a single die on a wafer and imaging that die with a topdown SEM or by using the data extracted using this approach as atraining data set for use with other non-destructive approaches.

Imaging and metrology of radially symmetric (e.g., circular) HARfeatures is described in detail in this specification to illustrate theinventive concepts, although those skilled in the art can extrapolatethe application of the disclosed technique to other geometries. Examplesof other geometries include trenches such as those used for shallowtrench isolation of transistors. As a non-limiting specific example, amemory hole of a 3DNAND structure is selected as a demonstrative exampleof a HAR feature to be synthesized for characterization. However, itshould be understood that the techniques described herein below withreference to the 3DNAND structure also apply to any other structure withHAR features.

FIG. 1 shows a portion of an example 3DNAND structure 100 with an etchedhard mask 110 on top of a multi-layer stack 120. The hard mask 110 canbe a carbon hard mask in an embodiment. The multi-layer stack can be anoxi-nitride (ON) stack in an embodiment. The top surface 115 of theetched hard mask 110 show the top openings of an array of memory holesetched into the structure 100. Each of the memory holes may be a HARfeature. The memory holes are seen more clearly in FIG. 3, which is aclose up of the portion 125 of the 3DNAND structure 100 shown within thedash-dotted outline. Note that the accompanying diagrams are notnecessarily drawn to scale, but are used to convey the inventive conceptvisually.

Following the etch process, the 3DNAND structure 100 may be sent to aFIB milling tool to mill a wedge-shaped test structure 200 thatlengthwise covers the entire depth of the memory holes through themulti-layer stack. Alternatively, another approach would be to mill acertain predetermined amount in the vertical direction with an ion-beammilling column and without even breaking vacuum move the sampleunderneath a SEM column and image the sample before going back to theion-beam column and repeating the process several times until the entiredepth of the HAR structure is traversed. The wedge-shaped test structure200 is illustrated in FIG. 2. Portions 205 shown with dashed lines areremoved to expose cross-sections (e.g., radial cross-sections) of thememory holes at different depths along the milled surface 215 of thewedge-shaped test structure 200. The milled memory holes with varyingdepths are seen more clearly in FIG. 4, which is a close up of theportion 225 of the wedge-shaped test structure 200 shown within thedash-dotted outline.

Note that during FIB milling, an aggressive ion flux can sometimesresult in a damage layer being formed on the top of the milled surface215 (the damage layer is not specifically shown for clarity). Thedamaged layer can be removed with another pass of the focused ion beamsat a lower fluence, if needed.

FIG. 3 shows a close up of the portion 125 (shown in FIG. 1) of the3DNAND structure 100 before milling, showing the memory holes 310 moreclearly. The memory holes 310 are substantively cylindrical features(i.e. having vertical channels 310 a with substantively uniform radialcross-section along the vertical direction, e.g. circular or ellipticalcross-sections) etched all the way through the hard mask 110 and themulti-layer stack 120. However, other HAR features may have othercross-sectional shapes. FIG. 4 shows a close up of the end portion 225of the wedge-shaped test structure 200 created by milling, showingexposed memory hole vertical channels 310 a with gradually varyingdepths along the lateral direction (x-direction).

FIG. 5 shows two different views 500A and 500B obtained by imaging thewedge-shaped test structure 200. Note that although 500A shows onecontinuous view, the view 500A is created by stitching a number ofrectangular fields of view (FOV) 550 of the SEM tool. Each rectangularFOV (e.g., 550 a, 550 b, 550 c, 550 d, 550 e, 550 f) is referred to as a“frame.” There can be some overlap between adjacent frames for seamlessstitching. For example, frames 550 a and 550 b overlap in the area 550 ab. Each memory hole 310 creates a corresponding elliptical image 510,when imaging is done normal to the milling plane (e.g., FIB plane BB′shown in FIG. 6).

View 500B shows reconstruction of a vertical section profile (along theplane AA′) of the wedge-shaped test structure 200. For a milling angleΦ, and a memory hole depth ‘h’, a swath length ‘L’ (where L=h/tan Φ)should be covered by the SEM to cover all the memory holes along the xdirection. In an example, for a 5 μm deep memory hole (the depthincludes about 800 nm of remaining hard mask on top of a multi-layeredstack), 5 or 6 FOVs or frames can be stitched to cover a swath length ata resolution of ˜2 nm, assuming a 20% overlap between the adjacentframes to obtain good stitching.

FIG. 6 shows a longitudinal two-dimensional shape 610 (in a verticalplane) representing a memory hole channel, which is not necessarilycylindrical. Typically, there is a bow in the memory hole caused bydeflected ions during the etch step following which there is aprogressive reduction in the diameter of the hole as the aspect ratioincreases. FIG. 6 also shows a projection of a radial cross-section ofshape 610 onto the wafer plane (i.e. a plane parallel to the plane ofthe substrate) as well as a projection of the same onto the millingplane (BB′). The milling plane is a plane along which the FIB (or othermilling tool) cuts off a portion of the structure 100 to create thewedge-shaped structure 200. A projection of a radial cross-section ofthe memory hole onto the wafer plane results in a circular image 612 fora top down imaging system with a conventional SEM, while a tilt SEM canobtain an elliptical image 614 (representing image projection onto themilling plane). In case of the later, a shear transformation should beapplied to the elliptical image 614 before further image-processing. Theshear transformation is a simple homogeneous transformation that is afunction of the incidence angle of the electron beams and is notrequired for the more common case of top-down imaging.

FIG. 7 shows a reconstructed 3D profile 700 of a memory hole. The 3Dprofile 700 is synthesized by combining different radial cross-sectionsat various depths (along z-direction) that are obtained from SEM imagingof different memory holes disposed along the x-direction (lateraldirection) of the wedge-shaped test structure 200. The 3D reconstructionalso illustrates one of the pitfalls of using a 2-dimensionalcross-section image such as that in FIG. 6. A significant asymmetry ofthe HAR feature, that can be readily observed from the 3D reconstructionof FIG. 7, is not evident in the 2D construction shown in FIG. 6 as aconsequence of the spatial aliasing that was referred to earlier.Details of the 3D profile reconstruction are elaborated further below.

FIGS. 8A-8D show four different radial cross-sections of a memory hole.These cross-sections are chosen to illustrate various shapes that couldresult from the etching process. Each one of these cross-sections hasdifferent harmonic content and a one-size-fits-all approach will lead topoor results. In each of the plots, the perfect circle is the idealradial cross-section of a memory hole at a given location, and thenon-circular enclosed curve is the statistical “best-fit” version of thereconstructed radial cross-section obtained from applying the methods ofthis disclosure to the SEM images, as described below.

FIG. 9 is a flow diagram of an example method 900 to reconstruct a 3Dprofile of a HAR feature, in accordance with some embodiments of thepresent disclosure. Some of the steps of method 900 can be performed byprocessing logic that can include hardware (e.g., processing device,circuitry, dedicated logic, programmable logic, microcode, hardware of adevice, integrated circuit, etc.), software (e.g., instructions run orexecuted on a processing device), or a combination thereof. Althoughshown in a particular sequence or order, unless otherwise specified, theorder of the processes in method 900 or other methods described belowwith illustrative flowcharts can be modified. Thus, the illustratedembodiments should be understood only as examples, and the illustratedprocesses can be performed in a different order, and some processes canbe performed in parallel. Additionally, one or more processes can beomitted in various embodiments. Thus, not all processes are required inevery embodiment. Other process flows are possible.

The method begins at block 910, where a wedge-shaped structure (such asstructure 200) is prepared with an array of HAR features (e.g., memoryholes) with varying depths. The wedge-shaped test structure can beprepared as described above.

At block 915, SEM images of the HAR features are obtained. In oneembodiment, the SEM images represent radial cross-sectional images ofthe HAR features obtained from the top surface 215 of the wedge-shapedstructure 200.

At block 920, edges to each cross-sectional image of a corresponding HARfeature are computed. This can be done using standard image processingroutines. One such non-limiting routine uses Sobel or Laplacianoperations with additional filtering and suppression. Canny edgedetection can also be used to remove speckle noise, detect edges andsuppress multiple edge detection in one step.

At block 925, the computed edges are radially re-sampled uniformlyaround a circumference of the image of the HAR feature. Number of samplepoints is user-selectable based on the geometry of the HAR feature, aswell as the target resolution.

At block 930, the re-sampled edges are represented as a set ofharmonics. For this operation, the edge pixels are expressed as afunction of angular position, and their spatial (x,y) coordinates aresaved. This is done for each HAR feature, e.g., for each memory hole.The spatial coordinates are then computationally transformed into thefrequency domain. This transformation can be done by using a FastFourier Transform (FFT) that computes a set of harmonics. For a closedgeometry (blob), the zero-th harmonic corresponds to the center of thebest-fit circle to the blob, the first harmonic corresponds to theradius of the best-fit circle, and the higher order terms illustrate thelack of circularity, i.e. evidence of striations. The formulae toillustrate this decomposition of the (x,y) coordinates of the edges inthe frequency domain are as follows:

x=x ₀ +r cos θ+Σ_(k=2) ^(n) a _(k) cos kθ

y=y ₀ +r sin θ+Σ_(k=2) ^(n) b _(k) sin kθ

Here, (x₀, y₀) is the center of the circle, r is the radius, and a_(k)and b_(k) are the coefficients of the higher-order harmonic terms. Inthe example shown in FIG. 8A, the center of the specific memory hole is(45 nm, 120 nm), the radius is 40 nm (corresponding to a criticaldimension (CD) of 80 nm which is equal to the diameter of the memoryhole), and the amplitude of the fifth harmonic is 8 nm. FIGS. 8A-8D showthe best fit to the ideal circle using different combinations ofhigher-order harmonics used to represent circularity of the HAR feature.

Existing methods define circularity based on a ratio of area and asquare of the perimeter (known as “area/perimeter² ratio”), or, a ratioof major axis and minor axis. Each of these definitions has theirlimitations. The major axis/minor axis ratio method only predictsaccurate circularity for perfectly elliptical geometries. Thearea/perimeter² ratio method loses information about local variationalong the edges. In this disclosure, circularity is defined as the ratiobetween the root mean square summation of higher order harmonic termsand the fundamental harmonic, i.e., radius.

At block 935, a plurality of radial cross-sectional profiles at varyingdepths are stitched to generate a composite 3D profile, such as profile700 shown in FIG. 7. The stitching is based on harmonic analysis ofimage data in the frequency domain instead of conventional imagestitching that is done in the spatial domain. FIG. 10 illustratesharmonic analysis of data obtained from a single slice after the FFT.The top curve in FIG. 10 shows post-FFT spectrogram amplitude of edgedata for the x-coordinate, and the bottom curve in FIG. 10 showspost-FFT spectrogram amplitude of edge data for the y-coordinate. Notethat edge data refers to raw image data available from the edge pixelsduring top-down SEM imaging. As mentioned above, some of the key metricsof the memory hole, such as location of the center, CD and circularity,can be computed from the harmonic information in the FFT. The plotsshown in FIG. 10 correspond to a memory hole with center at a (x,y)coordinate of (40 nm, 120 nm) and a radius of 40 nm (making the CD 80nm). FIG. 11 illustrates the spectrogram (as shown in FIG. 10) replottedwhen the zeroth harmonic (i.e. the center of the circle) and the firstharmonic (the radius of the circle) components are filtered out, showingmore prominently which higher-order harmonic term contributes the mostin correctly predicting the edge contour.

It should be noted that each image from the SEM corresponds to aparticular depth of the memory hole as exposed by the milling. As aresult, it is possible to assemble metrics for the center, CD andcircularity as a function of depth. These metrics along the depthdirection can be assembled to recreate the original memory hole profileor other HAR feature profile (e.g., a composite profile like what isshown in FIG. 7). Evolution of certain metrics (δx, CD and striationamplitude) with depth are plotted as two-dimensional charts in FIGS. 12and 13. The top curve in FIG. 12 shows the evolution of δx in thez-direction (δx being the deviation of the center of the memory holeform the ideal center in the x-direction), and the bottom curve showsthe evolution of CD in the z-direction. The plots in FIG. 12 illustratelinear tilt in the x-direction and a typically observable bow in suchHAR structures.

FIGS. 13, 14 and 15 plot real data from an etched memory hole. This datais obtained from the memory hole exposed at a certain depth because ofthe milling. Specifically, SEM images of a memory hole are obtained and“blobs” are detected using standard image processing algorithms for blobdetection. Edges are detected on the identified blobs and following aresampling process, FFTs are computed as a first step forharmonic-analysis based stitching. Following stitching, variousparameters are estimated for the various blobs. Histograms are assembledfor various groups of blobs (e.g., by image frame, by cell position, byz-depth) following which probability densities for geometric parameterssuch as CD and circularity are computed. The plots in FIGS. 13 and 14,respectively, show such probability densities for CD and circularitywhen the blobs are grouped by image frames. As noted above, circularityis defined as the ratio between the root mean square summation of higherorder harmonic terms and the fundamental zeroth order harmonic, i.e.,radius. The deviation of the centroid of the blobs from the targetposition at a certain depth is illustrated in FIG. 15. These deviationscan then be assembled together in the Z-direction to show the tilt ofthe various holes. The tilt of a memory hole varies spatially in the x-yplane, with the arrows in FIG. 15 representing both the magnitude anddirection of the tilt.

The nominal values of metrics (e.g., CD, circularity, and possiblyadditional metrics) as well as the statistical distributions for themetrics for all the memory holes in a Field of View (FOV) of the SEM areanalyzed. The process is repeated for multiple images, each imageobtained at a different depth.

The images are then assembled together and the combined data from allthe images is plotted as a function of depth, as shown in FIGS. 16-19.Specifically, FIG. 16 shows raw data obtained from individual blobs forCD, and FIG. 17 shows filtered data using smoothing operations (alsoreferred to as smoothening operations). The smoothing operation can usenon uniform rational B-splines (NURBs) to create seamless stitching ofimage data for different depths obtained from different FOVs or frameswhile preserving information about a radial cross-sectional profilecomputed using the higher-order harmonics, as described above. Thesmoothing operation can also be carried out using conventional filteringapproaches. Smoothing or filtering the harmonic coefficients in thez-direction is done to account for the hole-to-hole variability withinan array of memory holes in synthesizing one composite memory hole thatrepresents all the memory holes being considered. FIGS. 18 and 19 showboth the nominal value and the 1σ confidence interval (the shaded areaaround the plot, indicating standard deviation) of CD and circularity,respectively, plotted together to give a complete statistical picture ofthe performance of an etch recipe. In these plots, the dip (known as“necking”, shown within a dashed circle) at the interface of the hardmask and the multi-layer stack is prominently visible. As shown in FIG.19, circularity is almost held constant throughout the etching process.If there is any lack of circularity in the memory holes in the hardmask, that lack of circularity is propagated through the depth of themulti-layer stack.

Once the composite 3D profile is created, method 900 optionally proceedsto block 940. At block 940, the reconstructed HAR represented by the 3Dprofile is characterized, and the characteristic information is used totune a semiconductor process. For example, an etch process parameterscan be tuned to offset the effects of variation of etch rate withchanging aspect ratio.

FIG. 20 illustrates an example machine of a computer system 2000 withinwhich a set of instructions, for causing the machine to perform any oneor more of the methodologies discussed herein, can be executed. Inalternative implementations, the machine can be connected (e.g.,networked) to other machines in a LAN, an intranet, an extranet, and/orthe Internet. The machine can operate in the capacity of a server or aclient machine in client-server network environment, as a peer machinein a peer-to-peer (or distributed) network environment, or as a serveror a client machine in a cloud computing infrastructure or environment.

The machine can be a personal computer (PC), a tablet PC, a set-top box(STB), a web appliance, a server, a network router, a switch or bridge,or any machine capable of executing a set of instructions (sequential orotherwise) that specify actions to be taken by that machine. Further,while a single machine is illustrated, the term “machine” shall also betaken to include any collection of machines that individually or jointlyexecute a set (or multiple sets) of instructions to perform any one ormore of the methodologies discussed herein.

The example computer system 2000 includes a processing device 2002, amain memory 2004 (e.g., read-only memory (ROM), flash memory, dynamicrandom access memory (DRAM) such as synchronous DRAM (SDRAM) etc.), astatic memory 2006 (e.g., flash memory, static random access memory(SRAM), etc.), and a data storage device 2016, which communicate witheach other via a bus 2008.

The processing device 2002 comprises a graphics processing unit (GPU)for manipulating SEM images. In addition, the processing device 2002represents one or more general-purpose processing devices such as amicroprocessor, a central processing unit, or the like. Moreparticularly, the processing device can be complex instruction setcomputing (CISC) microprocessor, reduced instruction set computing(RISC) microprocessor, very long instruction word (VLIW) microprocessor,or processor implementing other instruction sets, or processorsimplementing a combination of instruction sets. Processing device 2002can also be one or more special-purpose processing devices such as anapplication specific integrated circuit (ASIC), a field programmablegate array (FPGA), a digital signal processor (DSP), network processor,or the like. The processing device 2002 is configured to executeinstructions for performing the operations and steps discussed herein.

The computer system 2000 can further include a network interface device2022 to communicate over the network 2018. The computer system 2000 alsocan include a video display unit 2010 (e.g., a liquid crystal display(LCD) or a cathode ray tube (CRT)), an alphanumeric input device 2012(e.g., a keyboard), a cursor control device 2014 (e.g., a mouse or atouch pad),), a signal generation device 2020 (e.g., a speaker), agraphics processing unit (not shown), video processing unit (not shown),and audio processing unit (not shown).

The data storage device 2016 can include a machine-readable storagemedium 2024 (also known as a computer-readable medium) on which isstored one or more sets of instructions or software embodying any one ormore of the methodologies or functions described herein. Theinstructions can also reside, completely or at least partially, withinthe main memory 2004 and/or within the processing device 2002 duringexecution thereof by the computer system 2000, the main memory 2004 andthe processing device 2002 also constituting machine-readable storagemedia.

In one implementation, the instructions include instructions toimplement functionality corresponding to a height differencedetermination. While the machine-readable storage medium 2024 is shownin an example implementation to be a single medium, the term“machine-readable storage medium” should be taken to include a singlemedium or multiple media (e.g., a centralized or distributed database,and/or associated caches and servers) that store the one or more sets ofinstructions. The term “machine-readable storage medium” shall also betaken to include any medium that is capable of storing or encoding a setof instructions for execution by the machine and that cause the machineto perform any one or more of the methodologies of the presentdisclosure. The term “machine-readable storage medium” shall accordinglybe taken to include, but not be limited to, solid-state memories,optical media and magnetic media.

Some portions of the preceding detailed descriptions have been presentedin terms of algorithms and symbolic representations of operations ondata bits within a computer memory. These algorithmic descriptions andrepresentations are the ways used by those skilled in the dataprocessing arts to most effectively convey the substance of their workto others skilled in the art. An algorithm is here, and generally,conceived to be a self-consistent sequence of operations leading to atargeted result. The operations are those requiring physicalmanipulations of physical quantities. Usually, though not necessarily,these quantities take the form of electrical or magnetic signals capableof being stored, combined, compared, and otherwise manipulated. It hasproven convenient at times, principally for reasons of common usage, torefer to these signals as bits, values, elements, symbols, characters,terms, numbers, or the like.

It should be borne in mind, however, that all of these and similar termsare to be associated with the appropriate physical quantities and aremerely convenient labels applied to these quantities. Unlessspecifically stated otherwise as apparent from the above discussion, itis appreciated that throughout the description, discussions utilizingterms such as “obtaining” or “determining” or “detecting” or“generating” or “representing” or “creating” or “using” or the like,refer to the action and processes of a computer system, or similarelectronic computing device, that manipulates and transforms datarepresented as physical (electronic) quantities within the computersystem's registers and memories into other data similarly represented asphysical quantities within the computer system memories or registers orother such information storage devices.

The present disclosure also relates to an apparatus for performing theoperations herein. This apparatus can be specially constructed for theintended purposes, or it can comprise a general purpose computerselectively activated or reconfigured by a computer program stored inthe computer. Such a computer program can be stored in a computerreadable storage medium, such as, but not limited to, any type of diskincluding floppy disks, optical disks, CD-ROMs, and magnetic-opticaldisks, read-only memories (ROMs), random access memories (RAMs), EPROMs,EEPROMs, magnetic or optical cards, or any type of media suitable forstoring electronic instructions, each coupled to a computer system bus.

The algorithms and displays presented herein are not inherently relatedto any particular computer or other apparatus. Various general purposesystems can be used with programs in accordance with the teachingsherein, or it can prove convenient to construct a more specializedapparatus to perform the method. The structure for a variety of thesesystems will appear as set forth in the description below. In addition,the present disclosure is not described with reference to any particularprogramming language. It will be appreciated that a variety ofprogramming languages can be used to implement the teachings of thedisclosure as described herein.

The present disclosure can be provided as a computer program product, orsoftware, that can include a machine-readable medium having storedthereon instructions, which can be used to program a computer system (orother electronic devices) to perform a process according to the presentdisclosure. A machine-readable medium includes any mechanism for storinginformation in a form readable by a machine (e.g., a computer). Forexample, a machine-readable (e.g., computer-readable) medium includes amachine (e.g., a computer) readable storage medium such as a read onlymemory (“ROM”), random access memory (“RAM”), magnetic disk storagemedia, optical storage media, flash memory devices, etc.

In the foregoing specification, implementations of the disclosure havebeen described with reference to specific example implementationsthereof. It will be evident that various modifications can be madethereto without departing from the broader spirit and scope ofimplementations of the disclosure as set forth in the following claims.The specification and drawings are, accordingly, to be regarded in anillustrative sense rather than a restrictive sense.

1. A computer-implemented method comprising: obtaining top-down scanningelectron microscope (SEM) images of cross-sections of an array ofhigh-aspect-ratio (HAR) features, wherein depths of the HAR features ofthe array gradually vary along a lateral direction from a maximum valueto a minimum value; detecting edges of the cross-sections of each HARfeature of the array of HAR features; re-sampling each of the detectededges in a spatial domain at a target angular resolution; representingeach of the re-sampled edges as a corresponding set of harmonics in afrequency domain, each set of harmonics preserving characteristicinformation about a respective cross-section of a HAR feature at acertain depth, wherein the set of harmonics is obtained by performingFast Fourier Transform (FFT) on the re-sampled edges; and generating acomposite three-dimensional (3D) profile of a HAR feature by stitching aplurality of cross-sections at various depths that are reconstructedfrom analyzing the corresponding sets of harmonics in the frequencydomain, wherein the stitching is based on the preserved characteristicinformation about each cross-section of the plurality of cross-sections.2. The method of claim 1, wherein the method further comprises: prior toobtaining the top-down SEM images, obtaining a coupon comprising thearray of high-aspect-ratio (HAR) features, each HAR feature of the arrayof HAR features having substantially identical initial depth; andpreparing a test structure for vertical sampling along a direction ofdepth of the HAR features by removing a portion of the coupon to createvariation in depth of the HAR features along the lateral direction. 3.The method of claim 2, wherein preparing the test structure for verticalsampling comprises: using a focused ion beam to create a wedge-shapedtest structure.
 4. The method of claim 2, wherein the maximum value isthe initial depth of the HAR feature, and the minimum value issubstantially close to zero.
 5. The method of claim 1, whereingenerating the composite three-dimensional profile of a HAR featurefurther comprises: smoothing out coefficients of the sets of harmonicsin a vertical direction to account for variability within the array ofHAR features, such that the composite three-dimensional profilerepresents the array of HAR features.
 6. The method of claim 5, whereinthe smoothing out of the 3D profile is performed using non-uniformrational B-splines.
 7. The method of claim 1, wherein the HAR feature isa vertical hole with a substantially circular radial cross-section. 8.The method of claim 7, wherein the preserved characteristic informationcomprises planar coordinates of a location of a center of the verticalhole being represented by a zero-eth order harmonic of the set ofharmonics.
 9. The method of claim 7, wherein the preservedcharacteristic information comprises a radius of the vertical hole beingrepresented by a first-order harmonic of the set of harmonics.
 10. Themethod of claim 7, wherein the preserved characteristic informationcomprises circularity of the vertical hole at a certain depth, thecircularity being represented by a ratio of a radius of the verticalhole and a summation of higher-order harmonic terms.
 11. The method ofclaim 1, wherein representing each of the re-sampled edges as acorresponding set of harmonics in a frequency domain comprises:performing Fast Fourier Transform (FFT) on the radially re-sampled edgesto generate the set of harmonics.
 12. The method of claim 11, where anoise analysis is performed to identify prominent orders of higher-orderharmonics from the set of harmonics.
 13. The method of claim 1, whereinthe method further comprises: obtaining device characterization datafrom the composite 3D profile of the HAR feature.
 14. The method ofclaim 13, wherein the method further comprises: tuning a semiconductorwafer processing recipe based on the device characterization data. 15.The method of claim 2, wherein the method further comprises: prior tore-sampling, applying shear transformation to the edges detected along atilted cutting plane created by removing a portion of the coupon
 16. Themethod of claim 1, wherein the target angular resolution of re-samplingis user-selectable based on a geometry of the HAR feature.
 17. A systemcomprising: a memory; and a processing device operatively coupled withthe memory, to: obtain top-down scanning electron microscope (SEM)images of cross-sections of an array of high-aspect-ratio (HAR)features, wherein depths of the HAR features of the array gradually varyalong a lateral direction from a maximum value to a minimum value;detect edges of the cross-sections of each of the HAR feature of thearray of HAR features; re-sample each of the detected edges in a spatialdomain at a target angular resolution; represent each of the re-samplededges as a corresponding set of harmonics in a frequency domain, eachset of harmonics preserving characteristic information about arespective cross-section of a HAR feature at a certain depth, whereinthe set of harmonics is obtained by performing Fast Fourier Transform(FFT) on the re-sampled edges; and generate a compositethree-dimensional (3D) profile of a HAR feature by stitching a pluralityof cross-sections at various depths that are reconstructed fromanalyzing the corresponding sets of harmonics in the frequency domain,wherein the stitching is based on the preserved characteristicinformation about each cross-section of the plurality of cross-sections.18. The system of claim 17, wherein the top-down scanning electronmicroscope (SEM) images are obtained from a test structure prepared forvertical sampling along a direction of depth of the HAR features. 19.The system of claim 17, wherein the processing device comprises agraphics processing unit.
 20. A non-transitory computer readable mediumcomprising instructions, which when executed by a processing device,cause the processing device to perform operations comprising: obtainingtop-down scanning electron microscope (SEM) images of cross-sections ofan array of high-aspect-ratio (HAR) features, wherein depths of the HARfeatures of the array gradually vary along a lateral direction from amaximum value to a minimum value; detecting edges of the cross-sectionsof each HAR feature of the array of HAR features; re-sampling each ofthe detected edges in a spatial domain at a target angular resolution;representing each of the re-sampled edges as a corresponding set ofharmonics in a frequency domain, each set of harmonics preservingcharacteristic information about a respective cross-section of a HARfeature at a certain depth, wherein the set of harmonics is obtained byperforming Fast Fourier Transform (FFT) on the re-sampled edges; andgenerating a composite three-dimensional (3D) profile of a HAR featureby stitching a plurality of cross-sections at various depths that arereconstructed from analyzing the corresponding sets of harmonics in thefrequency domain, wherein the stitching is based on the preservedcharacteristic information about each cross-section of the plurality ofcross-sections.